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3 THE MODULUS FUNCTION ON A GROUP        255

0,1, 2,..., iQ?  1), the residues mod p of the numbers z* are all distinct). Deduce
that in this case y~p. Hence, using the multiplicative property of norms of quater-
nions, deduce that every positive integer is the sum of at most four squares (Lagrange^s
theorem).

3.   (a)   Let A be Lebesgue measure on R. Let/be a real-valued function i>0 on R which
is A-integrable, bounded and of compact support. Put y sup/(0- For each weR,

r eR

let U/w) denote the set of t e R such that/(/) ^ w, and put vf(w) = A*(U/O)). Show
that, for all a > 1,

/+>                            -y

f*(t)dt=

J -oo                          Jo

-1 dw.

(b)    Let g be another function satisfying the same conditions as/, and put 3 = sup g(t).

reR

Let h be the function on R2 defined by h(u, v) =/O) + g(v) if f(u)g(v)^Q> and
h(u,v) = Q otherwise; also let k(t)= sup h(u,v), so that k is positive, A-integrable,

bounded and compactly supported. Show that, for all a > 1,

r+o                                       / 1     r + oo                       i    r + co                 \

fc dtZ(Y+Sr[\      f'(f) dt+j-\       g'(t) dt  .

J-oo                                 \y   J-oo                   Oj_oo              /

(Observe that if 0 < w ^ 1 we have \Jk(yw + Bw) ^ Uf(yw) -h U/(3H>), and use (a)
above and Problem 4(d) of Section 14.1).

(c)    Let An be Lebesgue measure on R", and let A, B be two An-integrable subsets of
R". Show that

((A)*(A + B))1'"  (An(A))1/n ~h (AnCB))1/"

(Brunn-Minkowski inequality). (Reduce to the case where A and B are compact. Then
use induction on n, Problem 4(d) of Section 14.1, the theorem of Lebesgue-Fubini,
the inequality established in (b) above, and Holder's inequality.)

4. Let p be a prime number. The normalized Haar measure p, on the compact group
Zp of p-adic integers (Section 12.9, Problem 4) is such that the measure of any closed
ball of radius p"~k is equal to p~k. (Show that Zp is the union of pk closed balls of radius
/?""*, no two of which intersect.)

3. THE MODULUS FUNCTION  ON A GROUP; THE MODULUS
OF AN AUTOMORPHISM

Let G be a locally compact group, ju a left Haar measure on G. For all
s, t e G we have

directly from the definitions (14.1.2), and therefore &(s)n is also a left-invariant
positive measure on G. Hence there is a unique real number AG(X) > 0 (also
written A(s)) such that $(S)JLI = AG(X)ju (14.1.5), and this number is clearly, by  (/(O dt)/t, is independent of the choice of interval